Dales, H.G. and Kania, Tomasz and Kochanek, Tomasz and Koszmider, Piotr and Laustsen, Niels (2013) Maximal left ideals of the Banach algebra of bounded operators on a Banach space. Studia Mathematica, 218 (3). pp. 245286. ISSN 00393223

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Abstract
We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinitedimensional Banach space E: (I) Does B(E) always contain a maximal left ideal which is not finitely generated? (II) Is every finitelygenerated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} for some nonzero x in E? Since the twosided ideal F(E) of finiterank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinitedimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitelygenerated, maximal left ideal of B(E) contains FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.
Item Type:  Journal Article 

Journal or Publication Title:  Studia Mathematica 
Uncontrolled Keywords:  /dk/atira/pure/researchoutput/libraryofcongress/qa 
Subjects:  
Departments:  Faculty of Science and Technology > Mathematics and Statistics 
ID Code:  66236 
Deposited By:  ep_importer_pure 
Deposited On:  17 Sep 2013 07:38 
Refereed?:  Yes 
Published?:  Published 
Last Modified:  20 Feb 2020 01:37 
URI:  https://eprints.lancs.ac.uk/id/eprint/66236 
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